gauss - Chapter 4 Gausss Law 4.1 Electric Flux . 4-2 4.2...

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Chapter 4 Gauss’s Law 4.1 Electric Flux. ......................................................................................................... 4-2 4.2 Gauss’s Law. ......................................................................................................... 4-3 Example 4.1: Infinitely Long Rod of Uniform Charge Density. ............................ 4-8 Example 4.2: Infinite Plane of Charge. ................................................................... 4-9 Example 4.3: Spherical Shell. ............................................................................... 4-12 Example 4.4: Non-Conducting Solid Sphere. ....................................................... 4-13 4.3 Conductors. ......................................................................................................... 4-15 Example 4.5: Conductor with Charge Inside a Cavity . ........................................ 4-18 Example 4.6: Electric Potential Due to a Spherical Shell. .................................... 4-19 4.4 Force on a Conductor. ......................................................................................... 4-22 4.5 Summary. ............................................................................................................ 4-23 4.6 Appendix: Tensions and Pressures . .................................................................... 4-24 Animation 4.1: Charged Particle Moving in a Constant Electric Field. .............. 4-25 Animation 4.2 : Charged Particle at Rest in a Time-Varying Field . .................... 4-27 Animation 4.3 : Like and Unlike Charges Hanging from Pendulums. ................. 4-28 4.7 Problem-Solving Strategies . ............................................................................... 4-29 4.8 Solved Problems . ................................................................................................ 4-31 4.8.1 Two Parallel Infinite Non-Conducting Planes. ............................................ 4-31 4.8.2 Electric Flux Through a Square Surface. ..................................................... 4-32 4.8.3 Gauss’s Law for Gravity. ............................................................................. 4-34 4.8.4 Electric Potential of a Uniformly Charged Sphere . ..................................... 4-34 4.9 Conceptual Questions . ........................................................................................ 4-36 4.10 Additional Problems . ........................................................................................ 4-36 4.10.1 Non-Conducting Solid Sphere with a Cavity. ............................................ 4-36 4.10.2 P-N Junction. .............................................................................................. 4-36 4.10.3 Sphere with Non-Uniform Charge Distribution . ....................................... 4-37 4.10.4 Thin Slab. ................................................................................................... 4-37 4.10.5 Electric Potential Energy of a Solid Sphere. .............................................. 4-38 4.10.6 Calculating Electric Field from Electrical Potential. ................................. 4-38 4-1
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Gauss’s Law 4.1 Electric Flux In Chapter 2 we showed that the strength of an electric field is proportional to the number of field lines per area. The number of electric field lines that penetrates a given surface is called an “electric flux,” which we denote as E Φ . The electric field can therefore be thought of as the number of lines per unit area. Figure 4.1.1 Electric field lines passing through a surface of area A . Consider the surface shown in Figure 4.1.1. Let ˆ A = An r be defined as the area vector having a magnitude of the area of the surface, , and pointing in the normal direction, . If the surface is placed in a uniform electric field E A ˆ n ur that points in the same direction as , i.e., perpendicular to the surface A , the flux through the surface is ˆ n ˆ E AE A Φ= ⋅ = ⋅ = EA En r r r (4.1.1) On the other hand, if the electric field E makes an angle θ with (Figure 4.1.2), the electric flux becomes ˆ n n cos E EA E A Φ= ⋅ = = EA r r (4.1.2) where is the component of E n ˆ E =⋅ En r r perpendicular to the surface. Figure 4.1.2 Electric field lines passing through a surface of area A whose normal makes an angle with the field. 4-2
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Note that with the definition for the normal vector , the electric flux is positive if the electric field lines are leaving the surface, and negative if entering the surface. ˆ n E Φ In general, a surface S can be curved and the electric field E ur may vary over the surface. We shall be interested in the case where the surface is closed . A closed surface is a surface which completely encloses a volume. In order to compute the electric flux, we divide the surface into a large number of infinitesimal area elements , as shown in Figure 4.1.3. Note that for a closed surface the unit vector is chosen to point in the outward normal direction.
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gauss - Chapter 4 Gausss Law 4.1 Electric Flux . 4-2 4.2...

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